Elsholtz 2016 egyptian fractions odd denominators

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corollary_1_2: States the doubly exponential lower bound exp(exp(c k / log k)) for the number of representations of 1 by k distinct odd unit fractions, k odd and large, as the P = 2 case of Theorem 1.1.

theorem_1_1: For squarefree P and k large (k odd when P is even), the number of representations of 1 by k distinct unit fractions with denominators congruent to plus or minus 1 modulo P is at least exp(exp(c(P) k / log k)).

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Christian Elsholtz, Egyptian fractions with odd denominators, The Quarterly Journal of Mathematics 67 (2016), no. 3, 425--430, doi:10.1093/qmath/haw020 (published online 28 June 2016); preprint arXiv:1606.02117.

The copy read for this card is the arXiv version v1 (7 June 2016, the only arXiv version), nine physical pages numbered 1--9: the arXiv:1606.02117v1 PDF (https://arxiv.org/pdf/1606.02117v1); 107,870 bytes. The journal version was not obtained and has not been compared with the preprint; the locators below are the preprint's page numbers and labels. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1606.02117), every other right reserved.

Read status: the statements consumed by the corpus, Theorem 1.1 and Corollary 1.2 (pp. 2--3), and the paper's restatement (1.2) of the earlier bounds (p. 2), were read clause by clause on the PDF pages (claims checked). The proof in Section 2 (pp. 3--7) was read for its structure and is summarized on the Theorem 1.1 page; no proof was rewritten and none has been independently reviewed.

Contents

• Section 1 (pp. 1--3) reviews the unrestricted count. With $\mathcal X_k=\{(x_1,\ldots,x_k):\sum 1/x_i=1,\ 0<x_1<\cdots<x_k\}$, display (1.2) on p. 2 restates the known bounds as $\exp(\exp(((\log2)(\log3)+o(1)),k/\log k))\le|\mathcal X_k|\le c_0^{(5/3+\varepsilon)2^{k-3}}$, crediting the lower bound to Konyagin and the upper bound to Browning and Elsholtz, with "$c_0=1.264\ldots$ is $\lim_{n\rightarrow\infty}u_n^{1/2^n}$, $u_n=1$ [sic], $u_{n+1}=u_n(u_n+1)$". One constant in this restatement does not match the primary source as its card records it: Konyagin's Theorem 1 has the factor $\tfrac13$ in the double exponent ($((\ln2)(\ln3)/3+o(1))\,n/\ln n$; see its page). No mismatch is found for $c_0$, though Browning and Elsholtz's paper itself was not read. The printed "$u_n=1$" does not say where the sequence $1,2,6,42,1806,\ldots$ starts, and from $u_1=1$ the limit $\lim u_n^{2^{-n}}$ is the printed $1.264\ldots$ (the Vardi constant). That is the normalization in which Elsholtz and Planitzer (arXiv:1805.02945v1, p. 1) state Browning and Elsholtz's bound, as $c_0^{(5/24+\varepsilon)2^k}$ with $c_0=1.264\ldots$ and $u_1=1$, which is the bound in (1.2), since $(5/3)2^{k-3}=(5/24)2^k$. From $u_0=1$, the indexing of Remark 3 of Elsholtz and Planitzer's 2021 paper, the limit is the square $1.5979\ldots$ instead. The section also recalls Sierpiński's existence result for odd denominators, the exact counts for $k=9$ (five solutions) and $k=11$ (379,118 solutions), and the earlier lower bound $\sqrt2^{\,k^2(1+o(1))}$ of Chen, Elsholtz and Jiang for odd denominators.
• Theorem 1.1 (pp. 2--3): for a squarefree $P=p_1\cdots p_s$ and $k$ sufficiently large ($k$ odd when $P$ is even), the number of representations $1=\sum_{i=1}^k1/x_i$ with distinct positive $x_i\equiv\pm1\pmod P$ is at least $\exp(\exp(c(P)\,k/\log k))$ for some $c(P)>0$.
• Corollary 1.2 (p. 3): the case $P=2$, distinct odd denominators and odd $k$. The paper adds (p. 3) that an upper bound of the form $\exp(\exp(c_2k))$ follows from the unrestricted bound (1.2); the abstract (p. 1) states both bounds for the odd-denominator count $S(k)$, $k$ odd.
• Section 2 (pp. 3--7): the proof, from a Bang--Zsigmondy divisor count (Lemma 2.1), Wigert's divisor bound (Lemma 2.2), van Albada and van Lint's theorem that every positive integer is a finite sum of distinct unit fractions from any arithmetic progression (Lemma 2.3, giving Lemma 2.4), a binary-expansion construction reaching a fraction $1/(P^t-1)$, and the divisor-splitting identity of Lemma 2.5. Remark 2.6 says the constant $c(P)$ was not worked out.

Compiled scope

Statements only. The construction is recorded as a sketch on the Theorem 1.1 page; the paper's lemmas and the parity argument for the necessity of odd $k$ when $P$ is even were read but not checked step by step. Nothing in the paper decides a catalog problem's status: for Problem 148 it gives an independent doubly exponential lower bound for a restricted count, which together with the monotonicity of the unrestricted count in $k$ bounds $F(k)$ from below with an unspecified constant.

Bears on. #148 (Theorem 1.1 at $P=2$, that is Corollary 1.2: a lower bound $\exp(\exp(ck/\log k))$ for the number of representations of $1$ by $k$ distinct odd unit fractions, $k$ odd and large, a subset of the solutions $F(k)$ counts; with Konyagin's monotonicity inequality it bounds $F(k)$ from below for all large $k$, with an unspecified constant).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.